Al Queen Sequence

We will establish a bijection between the number of triangles with integer sides and perimeter $n$ (Type 1), and the number of triangles with distinct integer sides and perimeter $n+6$ (Type 2).

Given a triangle in Type 1, let it have side lengths $a \leq b \leq c$ , satisfying the triangle inequality $a + b > c$ . We create a triangle in Type 2 of lengths $a + 1, b + 2, c + 3$ , satisfying $a+1 < b + 2 < c +3$ and perimeter is $a + b + c + 6$ . This also satisfies the triangle inequality since $(a+1) + (b+2) > c + 3$ .

Conversely, given a triangle in Type 2, let it have sides lengths $A < B < C$ , satisfying the triangle inequality $A + B > C$ . We create a triangle in Type 1 of lengths $A - 1, B - 2, C - 3$ satisfying $A - 1 \leq B - 2 \leq C - 3$ and perimeter $A + B + C - 6$ .

This establishes the bijection between these two sets.

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Note: The number of triangles with integer sides and perimeter $n$ is known as the Alcuin sequence , hence the name of this problem. It is given by the generating function

$\frac{ x^3 } { ( 1 - x^2 ) ( 1 - x^3 ) ( 1 - x^4) }.$